The Path Integral Quantization And The Construction Of The S-matrix In The Abelian And Non-Abelian Chern-Simons Theories

The covariant path integral quantization of the theory of the scalar and spinor particles interacting through the Abelian and non-Abelian pure Chern-Simons gauge fields is carried out and is shown to be mathematically ill defined due to the absence of the transverse components of these gauge fields. This is remedied by the introduction of the Maxwell or the Maxwell-type (in th non-abelian case) term which makes the theory superrenormalizable and guarantees its gauge-invariant regularization and renormalization . The generating functionals are constructed and shown to be formally the same as those of QED (or QCD) in 2+1 dimensions with the substitution of the Chern-Simons propagator for the photon (gluon) propagator. By constructin the propagator in the general case; the existence of two limits; pure Chern-Simons and QED (QCD) after renormalization is demonstrated. By carrying out carefully the path integral quantization of the non-Abelian Chern-Simons theories using the De Witt-Fadeev-Popov and the Batalin-Fradkin -Vilkovisky